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Projectile Motion Basics

Kinematics

Throw a ball and it follows a graceful curved path called a parabola. That curve is really two simple motions happening at once: steady motion sideways and accelerating motion up and down under gravity. Once you learn to split a projectile's motion into horizontal and vertical parts, you can predict where it will land, how high it will go, and how long it stays in the air.

Two motions in one

Projectile motion is the motion of an object thrown or projected into the air, with gravity as the only significant force acting on it. The key idea is to treat the horizontal and vertical motions separately. Horizontally, there is no acceleration (ignoring air resistance), so the horizontal velocity stays constant. Vertically, gravity accelerates the object downward at 9.81 m/s², so the vertical velocity changes steadily. The two motions are independent: the horizontal motion does not affect how long the object stays in the air, and the vertical motion does not affect how far sideways it travels per second.

Resolving the launch velocity

If a projectile is launched at speed v at an angle θ above the horizontal, its velocity splits into two components: horizontal **v_x = v cos θ** and vertical **v_y = v sin θ**. The vertical component decides the time of flight and the maximum height; the horizontal component decides how far it travels in that time. At the top of the flight, the vertical velocity is momentarily zero, but the horizontal velocity is unchanged and gravity is still acting. The time to reach the top is v_y / g, the maximum height is v_y² / (2g), and the range is v_x multiplied by the total time of flight.

Range, height and real life

For a projectile landing at the same height it was launched from, the range is greatest at a launch angle of 45 degrees, because that angle balances the horizontal and vertical components. Lower angles give a flatter, shorter flight; higher angles give a higher, longer-lasting flight but less range. In real life, air resistance shortens the range and makes the path slightly asymmetric, and a launch from a height, such as a cliff or a basketball player's hands, changes the landing time. Sports, fireworks and ballistics all depend on these trade-offs between angle, speed and height.

Common mistakes

• Using the full launch speed for the vertical motion. Correction: split the velocity first, only the vertical component v sin θ affects the height and flight time.
• Thinking horizontal motion affects the time of flight. Correction: time in the air depends only on the vertical motion and the heights involved.
• Assuming the velocity is zero at the top of the flight. Correction: only the vertical component is zero there, the horizontal component is unchanged.

Practice questions

1. A projectile is launched at 20 m/s at 30° above the horizontal. What is its initial vertical velocity?
   Show answer

   v_y = v sin θ = 20 x sin 30° = 20 x 0.5 = 10 m/s upward.

2. A ball is thrown horizontally at 8 m/s from a height of 5 m. How long does it take to hit the ground? Use g = 9.81 m/s².
   Show answer

   5 = 0.5 x 9.81 x t², so t² = 10 / 9.81 = 1.019, t = 1.01 s.

3. What launch angle gives the maximum range on flat ground (ignoring air resistance)?
   Show answer

   45 degrees.

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